1. Field of the Invention
The present invention relates to a model and accompanying algorithm to simulate and analyze ink ejection from a piezoelectric print head. More particularly, the model and algorithm of this invention include a quadrilateral-grid-based, selectively reduced bi-cubic interpolation scheme for re-distancing the level set in ink-jet simulations. The simulation model and algorithm may be embodied in software, hardware or combination thereof and may be implemented on a computer or other processor-controlled device.
2. Description of the Related Art
The level set method is a powerful technique for interface capturing. In the parent application, a rigorous level set projection algorithm on quadrilateral grids was disclosed. That algorithm has been used successfully on ink-jet simulations. The level set is usually initialized as “signed distances” to the interface. However, because the level set convection equation is solved in every time step to update the level set values, the level set usually does not remain as a signed distance function. As a result, the simulation should be stopped periodically and the level set re-distanced. To illustrate this point, consider the rising bubble problem. A fluid bubble is surrounded by a second heavier fluid. The bubble gradually rises and changes its shape due to gravity. The simulation result is plotted in FIGS. 1 (a)–(d), in which the interface (zero level set) is represented by solid lines and the ±0.1, 0.2, 0.3 levels are indicated by dashed lines. It can be seen that the level set gradually distorts as the simulation progresses from t=0 to t=0.4. The distortion of level set from a perfect distance function not only reduces simulation accuracy but also causes a stability problem.
Various algorithms can be applied to re-distance the level set. One idea is to use a standard contour plotting algorithm to determine the interface and then recalculate the level set values. FIGS. 2 (a)–(d) show the result of the rising bubble simulation with this kind of level set re-distancing being performed every ten time steps. It can be seen that the level set basically remains as a signed distance function. However, there are several new problems that arise from such linear re-distancing: (i) the interface may be moved during the re-distancing work; (ii) the work takes a lot of CPU time; and (iii) the work introduces a serious mass loss problem.